Monthly Archives: June 2014

Most, if not all, set theorists at one point or another were asked by a fellow mathematician to explain how forcing works. And many chose to give as an opening analogy field extensions. You can talk about how the construction of an algebraic closure is a bit similar, since the generic filter is a bit like the maximal ideal you use to make this construction; or you can talk about adding a transcendental number and the things that change as you add it.

But both these analogies would be wrong. They only take you so far, and not further. And if you wish to give a proper explanation to your listener, there will be no escape from the eventual logic and set theory of it all. I stopped, or at least I’m doing my best, using these analogies. I do, however, use the analogy of “How many roots does $x^{42}-2$ has?” as an example for everyday independence (none in $\mathbb Q$, two in $\mathbb R$ and many in $\mathbb C$). But this is to motivate a different part of the explanation: the use of models of set theory (e.g. “How can you add a real number??”, well how can you add a root to a polynomial?) and the fact that we don’t consider the universe per se. Of course, in a model of $\ZFC$ we can always construct the rest of mathematics internally, but this is not the issue now. Just like we have a model of one theory, we can have a model for another.

So why do we do it? Why do we keep explaining adding generic sets using the analogy of field extensions? Well, the easy answer is that field extensions are something that most mathematicians can understand pretty easily and it shows how we can “enlarge a universe”.

But here’s why we should stop doing that (and why I stopped doing that):

Most people think about field extensions as being subfields of the complex numbers. This allows for a particular fixed background universe from which we can draw the numbers that we add. In set theory it is easy to think that there is one universe of sets, and that we force over that universe, in which case, where did the generic set come from?

Moreover, if $F$ is an algebraically closed field, and we add a transcendental element to $F$, then there is a nice closure operation after which the resulting field has the same first-order theory as $F$ (and if $F$ is uncountable, then there is also an isomorphism between them). Short of miraculous black magic, I don’t know of any such example in the case of set theory. Once you extend a model by forcing, there’s no definable process to restore the theory of the model by enlarging the model without adding ordinals (the obvious case is $V=L$).

The analogy is not accurate, and we can do better. For example, we can consider actual generic objects. The term generic comes from topology (and to my knowledge, that is where it trickled into algebraic geometry as well; but please correct me if I’m wrong about that).

We say that a point $x$ is generic if it is an element of every dense open set. Generic objects are a lot like that. We have a partial order, it has a topology, and the generic real is something which lies in the intersection of all the dense sets from the ground model.

So we can talk about adding a generic, or considering a generic point. This way it’s also easy to explain why it has so many properties — each property happens on a dense open set, so the generic point must have it.

Or we can talk about what actually happens. We start with a countable model of set theory. There’s no shame in that. It’s like considering a countable field or another countable structure. Since it’s countable, it’s certainly not the collection of all sets. Be sure to explain why the model is only countable from the outside and not internally; and what does it mean that “the model thinks that …”, true it’s not that easy, but there’s a large payoff. You’re not lying anywhere.

Recently I watched a video of Richard Feynman being asked by a layman to explain why magnets behave the way they do. And Feynman said that’s a very good question, and proceeded for five minutes to give analogy of a curious alien which would ask all sort of questions that you and I would take for granted; and then he went to say that the reason he didn’t answer the question is that all the analogies he could make, or other people make, boil down to electromagnetic forces acting on a microscopic level. So if you would compare magnets to rubber, then the next reasonable question would be how does the rubber work, and by inquiring further you’d finally reach the original question again, how does the electromagnetic force work.

So Feynman didn’t want to deceive the layman, or confuse them with analogies which ultimately explain nothing. And I think that’s a wonderful approach when you try to teach someone something. If electromagnetic force is one of the fundamental forces, we have to take it the way it is, and we can’t explain it in simpler terms.

Similarly, if forcing is a technique which is in its own class in mathematics, then we can’t quite explain it in terms of other techniques. Every analogy would break down and cause confusion. In that case, maybe it is the simplest solution to just start right away from forcing, logic and set theory? Motivate what you do in terms of logic. We are interested in truth values of statements, this statement is a statement of the form “There exists a set such that …”, so we want to approximate this set, so by carefully choosing a set from outside the model our approximations are actualized in the new model.

One prominent set theorist once told me that an incredibly smart mathematician (from representation theory) once asked him to explain forcing to him. He began with the analogy of field extensions, and it seems to be fine, and as he continued he defined the generic extension. Now we want to examine what sort of sentences are true in that generic extension, and we have this magical theorem which tells us exactly when something is true in the generic extension.

Once the sentence “a formula in the language of forcing” was uttered, the eyes became vacant and the rest was moot. And neither of the mathematician is stupid, and the set theorist involved is a wonderful teacher.

So why didn’t it work? What can we learn from this? My guess is that the analogy sets a particular direction, and when you break from that analogy, it becomes confusing to the listener. If you weren’t prepared to hear the term “the language of forcing”, then you won’t be able to jump over that hurdle when you reach it. And any analogy to field extensions hides this hurdle from the listener.

It seems to me, if so, that there are plenty of reasons not to use analogies, and plenty of reasons to explain things as they are. And forcing is not a trivial idea, remember that it completely revolutionized set theory. So if your audience can’t grasp it over coffee, it’s not a big deal. Perhaps using broad strokes to paint a rough image of approximations is better in that case, or at least better than giving the wrong idea.